In order to broadcast transactions via the JSON-RPC they also need to be created through it. If you are looking to just broadcast any transactions you should take a look at Create a transactions for the public API.

{"jsonrpc":"2.0","id":"unique-request-id","method":"transactions.create","params":{"passphrase":"this is a top secret passphrase","amount":1000000000,"recipientId":"ANBkoGqWeTSiaEVgVzSKZd3jS7UWzv9PSo"}}

{"jsonrpc":"2.0","id":"unique-request-id","method":"transactions.bip38.create","params":{"bip38":"this is a top secret passphrase","userId":"123","amount":1000000000,"recipientId":"ANBkoGqWeTSiaEVgVzSKZd3jS7UWzv9PSo"}}